Graph convolution networks (GCNs) have become effective models for graph classification. Similar to many deep networks, GCNs are vulnerable to adversarial attacks on graph topology and node attributes. Recently, a number of effective attack and defense algorithms have been designed, but no certificate of robustness has been developed for GCN-based graph classification under topological perturbations with both local and global budgets. In this paper, we propose the first certificate for this problem. Our method is based on Lagrange dualization and convex envelope, which result in tight approximation bounds that are efficiently computable by dynamic programming. When used in conjunction with robust training, it allows an increased number of graphs to be certified as robust.

Appendix E provides a worked example of this.

Appendix E provides a worked example of this.

Convex optimization problems with staged structure appear in several contexts, including optimal control, verification of deep neural networks, and isotonic regression. Off-the-shelf solvers can solve these problems but may scale poorly. We develop a nonconvex reformulation designed to exploit this staged structure. Our reformulation has only simple bound constraints, enabling solution via projected gradient methods and their accelerated variants.

To prove Lemma 5.1, we just need to show that the Pontryagin's Maximum Principle for the auxiliary X, (S.3) and the following matrix trace properties: Tr(A) = Tr( A Since the above obtained PMP equations (S.2) are the same with the differential PMP in (13), we thus Based on Lemma 5.1 and its proof, we known that the PMP of the auxiliary control system, (S.2), is exactly the differential PMP equations (13). From (S.2c), we solve for U Proof by induction: (S.2d) shows that (S.8) holds for This completes the proof. 2 D Algorithms Details for Different Learning Modes SysID Mode, then use the learned dynamics as the initial guess in IRL/IOC Mode. In design of the quadrotor's control objective function, to achieve SE (3) maneuvering In Fig. S1, we show more detailed results of imitation loss versus iteration In Fig. S2, we show more detailed results of SysID loss versus iteration In Fig. S5, we use the On the cart-pole and robot-arm systems (in Figure 1a and Figure 1b), we learn a feedback policy by minimizing given control objective functions. In Fig. S3, we show the detailed results of control loss (i.e. the value of control objective S6, we have the following remarks. This can be seen in Fig. S3 and Fig. S6 (in Fig. S6, PDP results in a simulated trajectory which is closer to the optimal one than that This explains why PDP outperforms GPS in terms of having lower control cost (loss).

Many learning tasks can find their counterpart problems in control fields.

In revision, we will discuss other related topics like Bayesian models, MDP, and MPC.

This paper introduces Deep Statistical Solvers (DSS), a new class of trainable solvers for optimization problems, arising e.g., from system simulations. The key idea is to learn a solver that generalizes to a given distribution of problem instances. This is achieved by directly using as loss the objective function of the problem, as opposed to most previous Machine Learning based approaches, which mimic the solutions attained by an existing solver.